Absolutely monotonic function

Remarks

Example: $f(x) = e^{x}$ on $\R$ is absolutely monotonic, since $f^{(k)}(x) = e^x > 0$ for all $k$. Likewise $g(x) = e^{-x}$ is completely monotone, since $(-1)^k g^{(k)}(x) = e^{-x} > 0$. Non-example: $f(x) = x^3$ on $(-1, 1)$ fails at $k = 0$, since $f(-\tfrac{1}{2}) < 0$, although $f^{(3)} = 6 > 0$. The function $f$ is absolutely monotonic on $I$ if and only if $x \mapsto f(-x)$ is completely monotone on $-I$. On an interval that contains an endpoint, such as $[0, \infty)$, one asks the inequalities on the interior and continuity of $f$ at the endpoint.